numerical study of mass and heat transport in solid-oxide fuel cells

Chemical Engineering Science 62 (2007) 5473–5486www.elsevier.com/locate/ces

Numerical study of mass and heat transport in solid-oxidefuel cells running on humidified methane

Vinod M. Janardhanan, Olaf Deutschmann∗

Institute for Chemical Technology and Polymer Chemistry, University of Karlsruhe, Engesserstr 20, D-76131 Karlsruhe, Germany

Received 16 June 2006; received in revised form 18 January 2007; accepted 19 January 2007Available online 4 February 2007

Abstract

Internally reforming anode supported solid-oxide fuel cells (SOFC) running on humidified CH4 is studied numerically. The computationalframework employs detailed multi-step models for heterogeneous chemistry for Ni catalysts. The electrochemistry is implemented using amodified Butler–Volmer setting based on elementary charge transfer kinetics. Transport through the porous matrix is modeled using the dustygas model (DGM). Parameters required for the electrochemical model are adjusted to reproduce experimentally observed data. Obtainedparameters are then used to model a co-flow planar single cell under internally reforming conditions assuming the interconnect walls asadiabatic. Numerous runs with varying inlet conditions of cathode stream are carried out to deduce the behavior of temperature and currentdensity distribution in the cell. Results show that internal reforming generally leads to a temperature drop near the inlet boundary.� 2007 Elsevier Ltd. All rights reserved.

Keywords: SOFC; Internal reforming; Heat transfer; Heterogeneous chemistry

1. Introduction

Solid-oxide fuel cells (SOFCs) offer the possibility of cleanenergy production by utilizing a variety of fuels. Natural gasis considered as an ideal fuel for stationary SOFC applicationsdue to its wide spread availability and distribution infrastruc-ture. A range of hydrocarbons can be utilized in SOFC with orwithout upstream fuel processing. In the case of upstream fuelprocessing an external fuel processor can be used in front ofthe stack to produce a reformate fuel which is rich in synthesisgas (H2 and CO). Alternatively the reforming process can becarried out within the cell stack in two different manners: (1)indirect internal reforming and (2) direct internal reforming.In the case of indirect internal reforming the reforming unitcan be integrated to the SOFC stack, which further utilizes theheat generated by the cells. In case of direct internal reform-ing, the reforming process is carried out directly on the anodeof the cell which is catalytically active. However, if the oper-ating conditions and anode materials are not chosen carefully

∗ Corresponding author. Tel.: +49 721 6083138; fax: +49 721 6084805.E-mail address: [email protected] (O. Deutschmann).

0009-2509/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2007.01.043

the direct internal reforming of hydrocarbons (HC), especiallyfor higher HC, can lead to coke formation within the anodewhich can finally lead to the complete failure of the cell (Steele,1999; Finnerty et al., 1998). Certain recent studies reported thedirect electrochemical oxidation of HC using anode materialsresistant towards carbon formation (Murray et al., 1999; Parket al., 2000). Others have studied the effect of precious metaldopents on anodes to decrease the coking propensity (McIntoshet al., 2003; Takeguchi et al., 2003) and demonstrated the useof bimetallics in ceria based anodes to reduce coking (Leeet al., 2004). The present work focuses on the interactionsof heat transport, mass transport, heterogeneous chemistry,and electrochemistry under internal reforming conditions in aNi–YSZ anode supported cell using a quasi two-dimensionalcomputational framework.

Modeling the processes in a SOFC is a challenging taskdue to the coupled interactions of transport, chemistry, andelectrochemistry. Understanding the interactions of the aboveis critical for optimal cell design. There are several studieson the modeling of SOFCs, starting from simple polarizationmodels to stack based models. Each of these studies vary in theassumptions made and in the dimensionality of the problem.Chan et al. have reported studies based on simple analytical

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5474 V.M. Janardhanan, O. Deutschmann / Chemical Engineering Science 62 (2007) 5473–5486

Fig. 1. Finite volume representation of the geometry under consideration and the equation system solved for various components.

Table 1SOFC parameters

Parameters Values Units

AnodeThickness (la) 0.50 mmAverage pore radius (rp) 0.50 �mAverage particle diameter (dp) 2.50 �mSpecific area (As) 1025 cm−1

Porosity (�) 0.35Tortuosity (�) 3.80Charge-transfer coefficient (�a) 0.50

ElectrolyteThickness (le) 25.0 �m

CathodeThickness (lc) 30.0 �mAverage pore radius (rp) 0.50 �mAverage particle diameter (dp) 2.50 �mPorosity (�) 0.35Tortuosity (�) 3.80Charge-transfer coefficient (�a) 0.45

models (Chan et al., 2001, 2004; Chan and Xia, 2001). Zhuand Kee (2003) reported a general numerical model to describethe polarization effects, which can handle any fuel compositionand also the concept of direct use of HCs. In a previous studywe reported the isothermal operation of planar SOFC on pre-reformed fuel (Zhu et al., 2005). Keegan et al. (2002) reporteda stack level model for a cross flow planar SOFC. Habermanand Young (2004) and Costmagna et al. (2004) studied theoperation of integrated planar SOFC. Li et al. (Li et al., 2004;Li and Suzuki, 2004) and Hall (1999) have reported numericalstudies on tubular SOFC.

In the present theoretical study we report a mechanism basedon heterogeneous the non-isothermal operation of a co-flowplanar anode supported SOFC under direct internal reformingconditions. The approach utilizes the potential of on elementaryreactions for the steam reforming of methane on Ni catalysts.The following sections encompass the modeling frameworkadopted in this work. A schematic representation of the modelgeometry under consideration and the equation system solvedfor various components are shown in Fig. 1. The unit cell is

assumed to be anode supported with 500 �m anode and 5 cmin length. Other parameters concerning the geometry are givenin Table 1.

2. Model description

2.1. Fluid and energy transport

2.1.1. Channel flowFlow through fuel and air channels is assumed to be one

dimensional and laminar in nature. The plug flow equation forspecies continuity in the channels is given by

�(�f Yk)

�t= −�(�f �Yk)

�z+ Pe

Ac

JkWk, k = 1, . . . , Kg . (1)

The mass balance equation is used as

�m

�t= −�(m�)

�z+ �

Kg∑k=1

Pe

Ac

JkWk , (2)

where Pe is the perimeter associated with the membrane elec-trode assembly (MEA) and Ac is the cross sectional area ofthe channel. Assuming constant pressure in the channels thedensity is calculated from the ideal gas equation

pM = �f RT . (3)

In Eqs. (1) and (2) Jk is the flux at the electrode channel in-terface and is calculated using the dusty gas model (DGM)(Eq. (10)). Temperatures in the flow channels are determinedfrom the energy equation

�(�f Cpf Tf )

�t= − �(��f Cpf Tf )

�z− h

Hc

(Tf − Te)

+ h

Hc

(TI − Tf ). (4)

The first term on the right-hand side of Eq. (4) representsthe transport of energy due to the bulk fluid flow, the secondand third terms represent the heat transfer from the channel tothe electrode structure and the heat transferred from the inter-connect into the flow channels, respectively. The heat transfer

V.M. Janardhanan, O. Deutschmann / Chemical Engineering Science 62 (2007) 5473–5486 5475

coefficient h is evaluated from the Nusselt number

Nu = hDh

k, (5)

which is expressed empirically as (Hayes and Kolaczkowski,1997)

Nu = 3.095 + 8.933

(1000

Gz

)−0.5386

exp

(−6.7275

Gz

), (6)

where the Graetz number Gz is given by

Gz = Dh

zReP r , (7)

where Dh is the hydraulic diameter, z the axial position, Re theReynolds number, and Pr is the Prandtl number.

2.1.2. Porous media transportSpecies transport through the porous media is assumed to be

one dimensional across the thickness of the porous structureand is given by

�(��f Yk)

�t= −�(JkWk)

�y+ skWkAs , (8)

where sk is the molar heterogeneous production rate of thechemical species k and As is the specific catalyst area avail-able for surface reactions. The total density within the porousstructure can be calculated from

�(��f )

�t= −

Kg∑k=1

�(JkWk)

�y+

Kg∑k=1

skWkAs . (9)

In the above equations the fluxes Jk are evaluated using theDGM. According to DGM, the net species molar flux is givenby (Zhu et al., 2005)

Jk = −⎡⎣ Kg∑

l=1

DDGMkl ∇[Xl] +

⎛⎝ Kg∑

l=1

DDGMkl [Xl]De

l,kn

⎞⎠ Bg

�∇p

⎤⎦ .

(10)

The first term on the right-hand side of Eq. (10) representsthe diffusive flux and the second term represents the viscousflux. Though the viscous flux driven by the pressure gradientis negligible compared to diffusive flow in the porous anode(Jiang and Virkar, 2003), we retain the term in our calcula-tions. DDGM

kl are defined as DGM diffusion coefficients given as(Zhu et al., 2005)

DDGMkl = H−1, (11)

where the elements of the H matrix are

hkl =⎡⎣ 1

Dek,kn

+∑j �=k

Xj

Dekj

⎤⎦ �kl + (�kl − 1)

Xk

Dekl

. (12)

A detailed account of the DGM is given elsewhere (Jackson,1977; Mason and Malinauskas, 1983). The permeability Bg in

Eq. (10) is given by Kozeny–Carman relationship (Bear, 1972)

Bg = �3dp

72�(1 − �)2. (13)

Assuming the reaction heat is released on the solid surfacethe heat balance equation within the porous structure can bewritten as

�(�CpT )

�t= ∇ · (keff∇T ) + h

�y(Tf − T )

−Kg∑k=1

sWkAshk + Qr + Qe, (14)

where Cp is the specific heat capacity of the solid material.The first term on the right-hand side represents heat transferdue to conduction and the second term the heat transfer fromthe channel to the electrodes at the interface. The heat transferdue to gas diffusion is not accounted in the above equation,since it is not significant compared to the conduction in porousmedia. �y appearing in the second term results from the finitevolume integration over the discretized cells and must have afinite value. The fourth and fifth term on the right-hand side ofequation Eq. (14) represents the radiative heat transfer betweenthe electrode and the interconnect structure and the heat releasedue to electrochemical reactions, respectively. The radiationheat source is given by

Qr = 1

�y

[(T 4

I − T 4)

1/I + 1/ − 1

]. (15)

2.1.3. ElectrolyteThe global charge transfer reaction at the three-phase inter-

face can be written as

H2 + 1/2O2−2 → H2O + 2e−, (16)

however, only a part of the energy change during the reactionis released as heat, which amounts to

Qe = −T �Si

2F. (17)

Assuming the electrolyte to be one dimensional the heat balanceequation reduces to

�(�CpT )

�t= �

�z

(k�T

�z

)+ i2

e

. (18)

The last term in the above equation represents the ohmic heatingdue to ion transport in the electrolyte.

2.1.4. InterconnectThe heat balance equation for the interconnect can be sum-

marized as

�(�CpTI )

�t= �

�z

(k�TI

�z

)+ h

HI

(Tf − TI )

− 1

HI

[(T 4

I − T 4)

1/I + 1/ − 1

]. (19)


The first term on the right-hand side represents the conductionwithin the interconnect, the second and third terms represent theheat transferred to/from the channel to the interconnect and theradiation heat transfer to/from the interconnect to the electrodestructure. Eqs. (14), (18), and (19) require boundary conditionsat z = 0 and L which are given by

�T

�z

∣∣∣∣z=0

= 0,�T

�z

∣∣∣∣z=L

= 0. (20)

Modeling radiative heat transfer in a SOFC is a complex pro-cess, and the radiative transport within the semitransparent elec-trodes and electrolyte, and surface to surface heat transfer mustbe accounted for an accurate calculation. However, an exactknowledge of phenomenological properties like absorption co-efficient, refractive index, scattering coefficient, emissivity, re-flectivity, etc. are obstacles to model radiative heat transport inSOFC electrodes and electrolyte. Detailed discussions of heattransfer by radiation in SOFCs are given elsewhere (Damm andFederov, 2005a,b; Murthy and Federov, 2003; VanderSteen andPharoah, 2006). In an analysis of spectral radiation in SOFCelectrodes Damm and Federov (2005b) have proven that the ra-diation effects in SOFC electrodes are minimal and can safelybe neglected. Therefore, in the present work, only surface tosurface radiation is considered. Since the planar geometry con-sidered here is of high aspect ratio (L/d ≈ 50) the surfacescan be treated as black bodies with emissivity 1 (Damm andFederov, 2005a).

2.2. Heterogeneous chemistry

SOFCs can run on HC fuels without external reforming sincethe high temperature of operation allows the endothermic re-forming reactions (steam reforming and dry reforming, wherethe rate of steam reforming is higher than that of dry reform-ing) to produce H2 and CO. These electrochemically activespecies further participates in charge transfer chemistry pro-ducing H2O and CO2. The CO produced can also participate inwater–gas shift or Boudouard reaction depending on the stoi-chiometric availability of H2O (Clarke et al., 1997). The prod-ucts of charge transfer chemistry further replenishes reform-ing chemistry within the cell. Internal reforming eliminates theneed of an external reformer and the high air flow rate requiredfor cell cooling, resulting in reduced capital cost and increasedoverall efficiency of the system. Reforming kinetics of CH4on Ni catalysts are well studied and is given elsewhere (Dickset al., 2000; Xu and Froment, 1989). There are also reports onthe catalytic activity of YSZ (Zhu et al., 2004) which is themost commonly used ionic conductor in SOFC components.

Though the reforming reactions are endothermic in nature,the cell reactions and the various losses within the cell areexothermic. It is generally agreed that the cell reactions produceenough heat that can be utilized for carrying out the reformingreactions internally. Under typical operating conditions the heatrequirement for steam reforming is half the heat generated in thecell stack (Ahmed and Foger, 2000). Utilizing the cell generatedheat for endothermic reforming reactions is an issue which

is considered in detail in the results and discussion of thispaper.

In the present study we use a multi-step heterogeneous re-action mechanism, which is evaluated for Ni–YSZ anode sup-ports and in steam assisted catalytic partial oxidation of CH4in small channel monolithic reactors using Ni supported onalumina (Hecht et al., 2005). The reaction mechanism consistsof 42 reactions among six gas-phase species and 12 surfaceadsorbed species. More discussions about the mechanism aregiven elsewhere (Zhu et al., 2005; Hecht et al., 2005). How-ever, in this study we use a modified version of the mechanismwhich is thermodynamically consistent for a wider temperaturerange (Janardhanan and Deutschmann, 2006). This modifiedmechanism is also validated using the experiments reported byHecht et al. (2005). The limitations of the new mechanism arediscussed in Janardhanan and Deutschmann (2006). Compari-son against equilibrium predictions shows that the mechanismneeds further improvements concerning surface carbon deposi-tion at higher temperatures. However, the gas-phase composi-tions were in good agreement. Since in this work the feedstockis assumed to be CH4, and CH4 being the most stable HC, anypossibility of gas-phase reactions can safely be neglected. How-ever, for the case of higher HCs, significant gas-phase reactionscan happen in the fuel channel (Sheng and Dean, 2004).

2.3. Electrochemistry

It is well known that H2 and CO can participate in chargetransfer chemistry. However, the rate at which CO is convertedin water–gas shift by far exceeds the rate of electrochemicaloxidation of CO (Mastsusaki and Yasuda, 2000). When bothspecies are present in the system H2 dominates over CO in thecharge transfer chemistry and CO seems to play a minor roleand can safely be neglected (Sukeshini et al., 2006). Hence, inthe present study we assume that H2 is the only electrochem-ically active species. The charge transfer reactions occur atthe interface formed by the electrocatalyst, electrolyte and gas-phase boundaries, known as three-phase boundaries (Tanneret al., 1997). In the case of composite electrodes, this reactionzone can spread out into the electrode usually of the order offew ∼ 10 �m (Kim et al., 1999; Williford et al., 2003). This isa small fraction of the total thickness of the anode in case ofan anode supported cell and hence we can assume the chargetransfer reaction to be happening at a limited region near theelectrode–electrolyte interface. Therefore in our study we donot consider distributed charge transfer, but only the chargetransfer at the electrode–electrolyte interface.

The irreversibility associated with the cell processes leadsto various overpotential losses such as activation, ohmic, andconcentration losses. A detailed account of the various lossterms is given in Larminie and Dicks (2003). The operatingcell voltage is related to various losses as

Ecell = Erev − �a(i) − |�c(i)| − �ohm(i) − �conc(i), (21)

where �a and �c are the activation losses at the anode andcathode side, respectively, �ohm is the ohmic overpotential, and


�conc is the concentration overpotential. Since in this work weare modeling porous media transport the concentration overpo-tential is not treated explicitly. Erev is given by the well knownNernst equation

Erev = E0 + RT

2Fln

(pH2,ap

1/2O2,c

pH2O,a

). (22)

In the above equation E0 is the electromotive force (EMF) atstandard pressure. The reversible potential is calculated basedon the partial pressures of H2, O2, and H2O at the three-phaseinterface. The ohmic loss in Eq. (21) is defined as

�ohm = Rtoti, (23)

where Rtot is the total area specific resistance. In modern cells,however, the resistance caused by the anode and cathode ma-terials is negligible compared with the electrolyte resistance.Therefore in the following analysis only the resistance con-tributed by the electrolyte, i.e., Re is considered, which is de-fined as

Re = le

e

, (24)

where le is the thickness of the electrolyte, and e is the elec-trolyte conductivity defined as a function of temperature

e = 3.34 × 104 exp

(−10300

T

). (25)

The above equation is valid for YSZ electrolyte materialand gives an ionic conductivity of 2.26 S m−1 at 1073 K and0.086 S m−1 at 1273 K, which is consistent with the reportedvalues in Larminie and Dicks (2003).

The functional relation between the activation loses and cur-rent density is described by a modified Butler–Volmer formal-ism. For hydrogen oxidation this takes the form

i = i0

[exp

((1 + �a)F�a

RT

)− exp

(−�cF�a

RT

)]. (26)

The overall oxygen reduction reaction and the incorporationof the ions into the electrolyte can be written in Kröger–Vinknotation as

12 O2 + V••

o (el) + 2e−(c) ⇀↽ O×o (el). (27)

The above global reaction can be split into a number of steps.One of the possible sequence of mechanistic steps is describedbelow.

(1) Adsorption/desorption of oxygen

O2 + (c) + (c) ⇀↽ O(c) + O(c). (28)

(2) Surface diffusion to three-phase boundary (TPB) regions

O(c) ⇀↽ O(TPB). (29)

(3) Formation of O2− ions and the subsequent incorporationinto the electrolyte

O(TPB) + V••o (el) ⇀↽ O×

o (el) + (TPB). (30)

Assuming the charge transfer step 30 to be rate limiting, Zhuet al. (2005) derived the expressions for Butler–Volmer equation

i = i0

[exp

(�aF�c

RT

)− exp

(−�cF�c

RT

)], (31)

where i0 is the exchange current density and � is the asymmetryfactor. In general, the activation overpotential �c on the cathodeside is larger compared to the anode side �a .

In the present study the exchange current density is expressedas a function of temperature and the partial pressures of the re-actants and products participating in the charge transfer chem-istry. For hydrogen oxidation the exchange current density isgiven as

i0 = i∗H2

(pH2/p∗H2

)1/4(pH2O)3/4

1 + (pH2/p∗H2

)1/2, (32)

and for oxygen reduction

i0 = i∗O2

(pO2/p∗O2

)1/4

1 + (pO2/p∗O2

)1/2. (33)

Here i∗i is an adjustable parameter and is expressed as a functionof temperature

i∗i = ki exp

(− Ei

RT

). (34)

Expressions for p∗H2

and p∗O2

can be found in Zhu et al. (2005).

2.4. Solution procedure

Eqs. (1)–(4), (8), (9), (14), (18), and (19) form a systemof coupled non-linear equations, which can be written in theresidual form as

F(�) = 0, (35)

where the vector � is given by

� = [(T )ic, (Y, m, T )a, (Y, �, T , )a,1 . . . (Y, �, T , )a,n,

(T )e, (Y, �, T )c,1, . . . (Y, �, T )c,m, (Y, m, T )c, (T )ic]T.

(36)

To solve the equation system they are first cast into a finitevolume form. The channels, interconnects, and electrolyte aretreated as one dimensional and the reactor geometry is dis-cretized in the axial direction into 200 cells. The anode is dis-cretized into 25 cells along the thickness and the cathode into10 cells. The finite volume discretization allows to define thevariables at the cell centers or at cell faces. In our scheme thefluxes are defined at the cell faces, while all other properties


Fig. 2. Flowchart of solution algorithm for the equation systems.

are defined at the cell centers. In calculating the diffusive fluxJk at the channel–electrode interface the concentrations at theinterface are assumed to be the same as those in the bulk of thegas. The entire solution procedure follows a space marchingalgorithm: at each axial position the transient system of equa-tions are solved until a steady state solution is obtained. Theinitial condition at each axial position assumes the convergedsolution from the previous finite volume cell. Due to the ellip-tic nature of the heat balance equation in porous electrodes andthe conductive terms in the solid regions an outer iteration loopis formed around the marching algorithm. The equation systemis solved using the differential algebraic equation (DAE) solverLIMEX (Deuflhardt et al., 1987). The entire solution procedureconverges in few passes. A flow chart for the solution processis shown in Fig. 2.

The solution of the porous media problem also requires thecurrent density at the TPB. The current density is calculatedfrom the system of algebraic electrochemical model equations(21), (26), and (31). The unknowns can be written in the vectorform as

� = [i, �a, �c]T. (37)

This equation system is solved using a damped Newton itera-tion, which converges in three to four integrations.

The software is written in FORTRAN 77 and integratedinto the detailed chemistry software package DETCHEM(Deutschmann et al., 2004).

In the solution procedure adopted here, the button cell ex-periments reported by Liu and Barnett (2003) are simulatedto obtain realistic electrochemical model parameters. The pa-rameters thus obtained are further used for a planar unit cellsimulation. In this way we ensure that the model parametersused for single cell simulation are in concert with experimentaldata.

2.5. Results and discussion

In the examples discussed here we focus our attention on theperformance of humidified (∼ 3%H2O) methane entering a pla-nar SOFC at 800 ◦C and 0.3 m s−1. For the base case consideredhere, air is assumed to enter the cathode channel at 1.5 m s−1

and 650 ◦C. Button cell simulations are carried out to repro-duce the experimental data reported by Liu and Barnett (2003)and hence to deduce the electrochemical model parameters.The authors report high power densities for direct operation onmethane. The MEA structure consists of 0.5 mm Ni–YSZ anodewith 25 �m YSZ electrolyte and 30 �m LSM cathode. Table 1lists the parameters used to represent the MEA. The exchangecurrent parameters used to fit the experimental data are givenin Table 3. A detailed description of these button cell simula-tions is reported in Janardhanan and Deutschmann (2006). Inbrief the only parameters adjusted to reproduce the experimen-tal observations are the activation energy and pre-exponentialfactors in Eq. (34). The activation energies are found to be in


Fig. 3. Voltage and power density against current density. Comparison with experimental data (Liu and Barnett, 2003).

Electrolyte

1040.0995.0

Anode

Axial position (m)

Tem

per

atu

re(K

)H

eig

ht

(m)

Hei

gh

t(m

)

0.01 0.02 0.03 0.04

9901000101010201030

0.0001

0.0002

0.0003

0.0004

5E-061E-05

1.5E-052E-05

2.5E-05

1030.0990.0

Cathode

0.0 0.05

Fig. 4. Temperature profile within the membrane electrode assembly, top panel shows the anode, middle one the electrolyte and the bottom panel shows thecathode.

good agreement with the experiments on Ni patterned anodes(Bieberle, 2000). Comparison of simulated and reported exper-imental data is given in Fig. 3. The model parameters reason-ably reproduce the experimental observation. However, at thehighest temperature of 800 ◦C the maximum power density isunder predicted. In general the current–voltage curves are linearat high current densities. However, concavity of the curve in-creases with decreasing temperature indicating the dominanceof activation losses. Even at high temperature in the realm oflow current density activation losses are dominant. The agree-ment between the experimental observation and simulation datasupports the assumption of H2 as the electrochemically activespecies even for the case of HC rich feed.

Assuming the outer walls of the interconnects to be adiabatic,Figs. 4–6 illustrate the temperature profiles within the variouscomponents of the unit cell under consideration. Material prop-erties used for the simulations are given in Table 2. Parametersfor exchange current density, five in Table 3, are taken from

Janardhanan and Deutschmann (2006). Fig. 4 shows the tem-perature profiles within the MEA. The top panel of the figureshows the temperature profile within the anode, the middle onethe electrolyte and the bottom one the cathode. Within the MEAstructure the resulting temperature profile is the net effect ofheat absorbed or released as a result of heterogeneous chemicalreactions, heat release due to the electrochemical reaction at theTPB, resistive heating within the electrolyte, convective heattransfer to and from the channels, and the radiative heat transferwith the interconnects. Near the inlet section, the temperaturedrops as a result of endothermic reforming reaction. However,further downstream the unit cell the temperature start to in-crease as a result of exothermic cell reactions. It is evident fromFig. 6 that the fuel stream looses heat to the comparatively coldair entering the air channel. As a result the temperature of the airstream increases from 650 ◦C to approximately 680 ◦C near theinlet (Fig. 5). However, as reforming starts the air channel tem-perature starts to drop. Both air and fuel channel temperature


Fig. 5. Temperature profile within the air channel and cathode side interconnect along the length of unit cell.

Fig. 6. Temperature profiles within the fuel channel and the anode side interconnect along the length of the unit cell.

Table 2Material properties

Parameters Values Units

InterconnectThickness (li ) 300 �mSpecific heat (Cp) 550.0 J kg−1 K−1

Thermal conductivity (k) 20.0 J m−1 s−1 K−1

Density (�) 3030.0 kg m−3

ElectrolyteSpecific heat (Cp) 470.0 J kg−1 K−1

Thermal conductivity (k) 2.16 J m−1s−1 K−1


CathodeSpecific heat (Cp) 430.0 J kg−1 K−1



AnodeSpecific heat (Cp) 450.0 J kg−1 K−1



start to increase further down the channel length due to theheat liberated within the MEA structure. Both Figs. 5 and 6show the temperature of the interconnect structures. There isno significant difference in temperature between the anode sideinterconnect and the fuel channel temperature. However, there

Table 3Parameters for exchange current density

Parameters Value Units

H2 oxidation (i∗H2)

kH2 (A cm−2) 207 × 103 A cm−2

EH2 (kJ mol−1) 87.8 kJ mol−1

O2 reduction (i∗O2)

kO2 (A cm−2) 51.9 × 103 A cm−2

EO2 (kJ mol−1) 88.6 kJ mol−1

is a significant temperature difference between the air streamand the cathode side interconnect. The electrolyte temperaturebegins to increase near the inlet mainly due to the heat gainedfrom the anode side fuel as well as the resistive heating withinthe electrolyte. It should be noted that in reality within a cellstack the temperature boundary condition for a unit cell is gov-erned by the surrounding cell temperatures. Furthermore eachunit cell’s temperature also depends on its position within thestack. However, an analysis like the one carried out here helpsto understand the possible temperature distribution and hencethe thermal miss-matches within a unit cell as a result of directinternal reforming of any hydrocarbon stream.

Fig. 7 shows the current density and the reversible potentialfor the unit cell operating at a cell potential of 0.7 V. As ex-pected the reversible potential peaks near the channel inlet anddrops further down the length. The initial increase in reversiblepotential results from the drop in temperature as a result of


Fig. 7. Current density and reversible potential as a function of reactor length with Ecell = 0.7 V.

Fig. 8. Over potential losses as a function of reactor length.

reforming reactions. An increase of open circuit potential withdecreasing temperature is typical for hydrogen oxidation mech-anism, which is the basic assumption for the electrochemicalmodel adopted here. However, down the unit cell length thereversible potential drops due to the increase in temperatureas well as the increased fuel dilution due to the production ofH2O at the TPB. The current density begins to increase nearthe inlet, however, starts to decrease as temperature drops andas the temperature increases the current density also increases.As long as enough fuel is present in the system which canbe electrochemically oxidized (in this case H2), the currentdensity depends strongly on the temperature than on the fueldilution effect. Fuel dilution at constant temperature would def-initely bring down the current density. However, for the caseconsidered here, down the length of the unit cell the tempera-ture increases continuously and the fuel composition becomesmore and more rich in H2 which is assumed to be the electro-chemically active species (Fig. 9). Fig. 8 shows various lossesas a function of position along the unit cell. As expected thecathode side losses by far exceed the anode side losses, whichis quite typical for SOFCs.

The variations in the species composition in the fuel channelas a result of chemical and electrochemical reactions in theMEA are illustrated in Fig. 9. All reaction products increasedown the channel. CH4 undergoes heterogeneous reformingwithin the anode producing H2. The H2 so produced furtherparticipates in the charge transfer chemistry at the three-phaseinterfaces producing more H2O. It should be noticed that H2O

is solely a result of electrochemical reactions. The electro-chemically produced H2O can further participate in reformingto produce synthesis gas as well as in water–gas shift reactions.The CO2 produced form the water–gas shift can also partici-pate in reforming and hence contribute to produce more H2. Itshould be noticed that the off gas of the cell is still a CH4 richfuel. CH4 conversion for the case presented here is 51.3%.The fuel utilization can further be increased by increasing theresidence time.

Figs. 10–12 show the species profiles and the surface cover-ages within the anode at three different positions along the unitcell length. Fig. 10 shows the conditions near the inlet, Fig. 11that at the middle of the cell, and Fig. 12 at the cell exit. In allcases H2 and H2O show opposite fluxes, with the flux of H2towards the TPB where it is consumed electrochemically andH2O away from the TPB where it is produced. Similarly CH4also has a flux into the anode where it undergoes steam reform-ing or dry reforming with CO2. It is quite interesting to noticethat the profiles are very much non-linear near the channel in-let, while showing linear behavior in other cases. All the threefigures also show the surface converges of various adsorbedspecies. In all cases, CO and H2 were found to be the majorspecies covering the surface. Near the channel inlet Ni remainsrelatively open in the vicinity of anode channel interface andmore CO starts to cover the surface near the TPB. From themiddle section onwards the surface coverages of hydrogen andCO remain nearly constant, but covering more surface than thatwas found near the inlet section.


Fig. 9. Mole fraction of various species in the fuel channel along the length of the reactor.

Fig. 10. Species profiles and surface coverages within the anode near the inlet section (z = 0.125 mm). Top panel shows the surface coverage and bottompanel shows species profiles.

Understanding the processes near the TPB would be the keyto further improvement of SOFC performance. The surface cov-erages of various reaction intermediates near the TPB can playa significant role in determining the rate limiting step. Fig. 13shows the surface coverages of hydrogen, CO and the uncov-ered Ni surfaces along the TPB. CO is found to be the speciescovering the major part of the surface, with the coverage in-creasing from the inlet section and slightly falling off near theexit. Surface coverage of hydrogen is plotted against the Y1axis. As observed in the gas-phase hydrogen coverage increasesfrom the inlet towards the exit. Though the oxygen coveragesis very small it is nevertheless found to be present the TPB andis plotted against Y2 axis. It should be noticed that oxygen onthe surface mainly results from the dissociative adsorption ofH2O. Oxygen coverage decreases initially and increase again

as more and more H2O is produced. The presence of reactionintermediates in the immediate vicinity of the TPB can reducethe availability of free surface for H2 adsorption and the sub-sequent electrochemical oxidation. Surface diffusion of the ad-sorbed species can also play a role in deciding the rate limitingstep, at least in the high current region (Williford et al., 2003).It is quite evident from Fig. 13 that open sites (Ni(s)) are maxi-mum in the region of low current (near the inlet) where the de-mand for TPB is low. Though the mechanism is not thoroughlyvalidated for the case of carbon formation, Fig. 14 shows qual-itative information regarding the carbon deposition within thecell along the TPB. As observed in experiments a higher cur-rent density is found to mitigate coke formation. Fig. 15 showsthe profile of oxygen along the channel. Near the inlet the rateof O2 consumption is found to have a maximum due to the


Fig. 11. Species profiles and surface coverages within the anode at half way from the inlet (z = 2.5125 cm). Top panel shows the surface coverage and bottompanel shows species profiles.

Fig. 12. Species profiles and surface coverages within the anode near the reactor exit (z = 4.9875 cm). Top panel shows the surface coverage and bottom panelshows species profiles.


Fig. 13. Surface coverages of the major surface adsorbed species at the three-phase boundary along the length.

Fig. 14. Surface coverages of carbon along the three-phase boundary.

Fig. 15. Mole fraction of oxygen in the air channel along the length of the reactor.

increase in current density. For the case considered here O2utilization is 26%.

To understand the behavior of temperature and current den-sity distribution on air inlet conditions, simulations are carriedout with the following conditions. (a) at 650 ◦C and 3 m s−1

(b) 800 ◦C and 1.5 m s−1. In call cases the current density, tem-perature and species profiles were found to follow the samepattern. A higher air inlet temperature was found to result inhigher current density. Various values for the base case andcases (a) and (b) are given in Table. 4. The table lists the av-erage value of current density, reversible potential, activationand ohmic losses, CH4 conversion and O2 utilization, and theexit temperatures of the fuel and air channel, respectively.

2.6. Conclusions

We have studied the operation of a planar SOFC using aCH4 rich fuel under internal reforming conditions. The elec-trochemical parameters used in the model well reproduce theexperimental data. It was found that the direct internal reform-ing can lead to a drop in cell temperature near the inlet and,hence utilizing the cell-produced heat for reforming reaction isnot possible in the case of co-flow configuration because thecell generates heat further down the channel. A counter flowconfiguration would be more appropriate for utilizing the cellgenerated heat for the endothermic reforming reactions, whichwill be described in a forthcoming paper. Effects of air inlet


Table 4Effect of air inlet conditions on electrochemical model parameters and other fuel and air channel variables

Case i (A cm−2) Erev (V) �a (V) �c (V) �ohm (V) CH4 (%) O2 (%) Tf (K) Tair (K)

Base 0.8387 0.9831 0.0448 0.1541 0.0827 55.19 26.3 1080.3 958.5(a) 0.7815 0.9896 0.0438 0.1626 0.0832 53.9 12.5 1074.2 942.8(b) 1.2024 0.9527 0.0521 0.1243 0.0762 66.5 40.5 1145.3 1074.2

conditions have been studied by keeping the fuel inlet condi-tions constant. It was found that a higher air inlet can resultin higher current density and higher waste heat production. Aqualitative study of coke formation shows that in the case ofinternal reforming coking can occur near the reactor inlet. Thefuel composition considered here resulted in a hydrogen richoff gas for a 5 cm long channel. Anode gas recirculation wouldbe an attractive option in such cases or even a longer fuel cellreactor would be much more efficient from the point of viewof fuel utilization.

Notation

Ac area of cross section, m2

As specific area, m−1

Cp specific heat, J kg−1 K−1

Dekl effective binary diffusion, m2 s−1

Del,kn effective Knudsen diffusion, m2 s−1

Dh hydraulic diameter, mE activation energy, J mol−1

Ecell operating cell potential, VErev Nernst potential, VF Faraday’s number, A mol−1

Gz Graetz numberh heat transfer coefficient, J m2 s−1 K−1

Hc channel height, mi current density, A cm−2

i0 exchange current density, A cm−2

Jk species fluxes, mol m−2s−1

k thermal conductivity, J m−1 s−1 K−1

K number of gas-phase speciesm mass flux, Kg m−2 s−1

M average Molecular mass, Kg mol−1

Nu Nusselt numberp pressure, PaPe perimeter, mPr Prandtl numberR gas constant, J mol−1 K−1

Re Reynolds numbersk rate, mol m−2 s−1

S entropy, J mol−1 K−1

T temperature, Ku velocity, m s−1

Wk molecular weight, Kg mol−1

[Xk] species concentration, mol m−3

X mole fractionYk mass fraction

Greek letters

� symmetry factor� Kronecker delta emissivity�conc concentration overpotential, V�a, �c activation overpotential anode, cathode, V�ohm ohmic concentration overpotential, V� viscosity, kg m−1 s−1

� stoichiometric coefficient reactants�′ stoichiometric coefficient products� density, kg m−3

Stefan Boltzmann constant, W m−2 K−4

e conductivity, S m−1

� porosity� cite density, mol cm−2

Subscript

a anodec cathodee electrolytef fluidg gasI, ic interconnectk species index

Abbreviation

DGM dusty gas model

Acknowledgments

We gratefully acknowledge many fruitful discussions withProfessor R.J. Kee and Dr. Huyang Zhu (Colorado School ofMines).

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